[EM] Re: Chain Climbing --> Chain Filling
heitzig-j at web.de
Sun Mar 13 13:52:10 PST 2005
> Thanks to both of your responses, I have an idea now that I think will work,
> and it should have (my) desired quality of encouraging generous approval
> cutoff and ranking of candidates below the cutoff.
> Basically, the idea is simply Beatpath: Break each cycle at the weakest link.
> But what should be the weakest link? Why not call it the defeat made by the
> candidate with lowest approval? We could call this Total Approval
> Beatpath (TAB), but suggest a better name if you want.
I now remember what conclusion I came to about this idea when I first
thought about it some months ago: It will always elect the least
approved candidate which beats all more approved ones. Proof: (i) The
winner, say X, beats all more approved ones; assume otherwise that Y
beats X and has more approval; then the beatpath Y>X is stronger than
all beatpaths X>... . (ii) No candidate with lower approval than X can
beat all more approved candidates; assume otherwise that Y beats all
more approved candidates and has less approval; then the beatpath Y>X is
stronger than all beatpaths ...>Y. This proof applies not only to
Beatpath with your definition of defeat strength but to all immune cycle
breaking methods (Beatpath, River, Ranked Pairs, ...) with your
definition of defeat strength.
In other words, this is the same method as Kevin's!
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